What does range mean?
In functions, the range is the set of all possible output values. For f(x) = |x|, the absolute value ensures the output is never negative. No matter which real number you input, the output is zero or positive. Therefore, the range of f(x) = |x| is all non-negative real numbers, expressed in interval notation as [0, ∞).
Key definitions: domain versus range
Domain and range basics
Domain refers to all valid inputs a function can accept; for f(x) = |x|, the domain is all real numbers because absolute value is defined everywhere. Range refers to all resulting outputs produced by those inputs. Because |x| converts negatives to positives and leaves non-negatives unchanged, the smallest possible output is 0, and outputs can grow indefinitely. This establishes the range as [0, ∞).
Visual verification with the graph
The graph of f(x) = |x| is a V shape with its vertex at the origin (0, 0). The arms extend upward infinitely, confirming that y values start at 0 and increase without bound. No part of the graph lies below the x-axis, visually confirming that negative outputs are impossible and that zero is the minimum value in the range.
Interpreting the graph for range
- The vertex at (0, 0) shows the minimum y-value is 0.
- As x moves away from 0 in either direction, y increases without limit.
- The graph exists only at y ≥ 0, aligning with [0, ∞) as the range.
Algebraic reasoning
By definition, |x| equals x when x ≥ 0, and −x when x
Testing key inputs
| Input (x) | Output f(x) = |x| | Explanation |
|---|---|---|
| 0 | 0 | Zero input yields zero output, the smallest value. |
| -3 | 3 | Negative input produces a positive output. |
| 5 | 5 | Positive input returns the same positive output. |
| -100 | 100 | Large negative input yields a large positive output. |
| 1000 | 1000 | Large positive input yields a large positive output. |
Common misconceptions
Some assume the range includes negative numbers because x can be negative. However, the absolute value operation changes negatives to positives, so outputs stay non-negative. Another misconception is that the range is limited to integers; in fact, it includes all non-negative real numbers, such as 0.5 and π, because inputs like ±0.5 and ±π produce those outputs.
Formal notation and summary
Using set builder notation, the range can be written as { y ∈ ℝ | y ≥ 0 }. In inequality form, this is y ≥ 0. The range of f(x) = |x| is therefore all real numbers greater than or equal to zero, with no upper bound. This conclusion holds for all real inputs and is foundational for understanding more advanced concepts such as inverse relations and absolute value equations.